# Subgroup of abelian group not implies powering-invariant

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup of abelian group) neednotsatisfy the second subgroup property (i.e., powering-invariant subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about subgroup of abelian group|Get more facts about powering-invariant subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property subgroup of abelian group but not powering-invariant subgroup|View examples of subgroups satisfying property subgroup of abelian group and powering-invariant subgroup

## Statement

It is possible to have an abelian group and a subgroup of that is not a powering-invariant subgroup of : there exists a prime number such that is powered over , but is not powered over .

## Related facts

Thus, our example must be an infinite subgroup of infinite index.

## Proof

The simplest example is as follows: is the additive group of rational numbers and . In this case, is powered over all primes, but is not powered over any.